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Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

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Abstract

In this paper, we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt \((0<q<1)\). This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to \({m} (m \leqslant 2)\), the global accuracy of k level corrected approximation is \(O(N^{-(2m(k+1)-\varepsilon)})\), where N is the number of the nodes, and \(\varepsilon\) is an arbitrary small positive number.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematical and System Sciences, Chinese Academy of Sciences, No.55, Zhongguancun East Road, Haidian District, Beijing, 100190, China

    Junmin Xiao & Qiya Hu

Authors
  1. Junmin Xiao
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  2. Qiya Hu
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Corresponding author

Correspondence to Qiya Hu.

Additional information

Communicated by: Kendall Atkinson

This work is supported by the Major Research Plan of Natural Science Foundation of China G91130015, the Key Project of Natural Science Foundation of China G11031006 and the National Basic Research Program of China G2011309702.

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Xiao, J., Hu, Q. Multilevel correction for collocation solutions of Volterra integral equations with proportional delays. Adv Comput Math 39, 611–644 (2013). https://doi.org/10.1007/s10444-013-9294-3

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